How to Calculate Inverse of 3×3 Matrix

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3×3 Matrix Inverse Calculator :root { –primary-blue: #004a99; –success-green: #28a745; –light-background: #f8f9fa; –dark-text: #333; –border-color: #ddd; } body { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: var(–light-background); color: var(–dark-text); line-height: 1.6; margin: 0; padding: 20px; } .calculator-container { max-width: 800px; margin: 20px auto; background-color: #fff; padding: 30px; border-radius: 8px; box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1); } h1, h2 { color: var(–primary-blue); text-align: center; margin-bottom: 20px; } .input-section, .result-section, .article-section { margin-bottom: 30px; padding: 20px; border: 1px solid var(–border-color); border-radius: 6px; background-color: #fff; } .input-grid { display: grid; grid-template-columns: repeat(3, 1fr); gap: 15px; margin-bottom: 20px; } .input-group { display: flex; flex-direction: column; } .input-group label { margin-bottom: 5px; font-weight: bold; color: var(–primary-blue); } .input-group input[type="number"] { padding: 10px; border: 1px solid var(–border-color); border-radius: 4px; font-size: 1rem; width: 100%; box-sizing: border-box; } .input-group input[type="number"]:focus { outline: none; border-color: var(–primary-blue); box-shadow: 0 0 5px rgba(0, 74, 153, 0.3); } .button-container { text-align: center; margin-bottom: 20px; } button { background-color: var(–primary-blue); color: white; padding: 12px 25px; border: none; border-radius: 5px; font-size: 1.1rem; cursor: pointer; transition: background-color 0.3s ease, transform 0.2s ease; } button:hover { background-color: #003366; transform: translateY(-2px); } button:active { transform: translateY(0); } #result { background-color: var(–success-green); color: white; padding: 20px; text-align: center; font-size: 1.5rem; font-weight: bold; border-radius: 5px; word-break: break-all; /* To handle long matrix output */ min-height: 60px; display: flex; align-items: center; justify-content: center; } #result pre { margin: 0; white-space: pre-wrap; word-wrap: break-word; } .error-message { color: #dc3545; font-weight: bold; margin-top: 15px; text-align: center; } .article-section h2 { text-align: left; margin-bottom: 15px; } .article-section p, .article-section li { margin-bottom: 15px; } .article-section code { background-color: var(–border-color); padding: 3px 6px; border-radius: 3px; font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace; } /* Responsive adjustments */ @media (max-width: 768px) { .calculator-container { padding: 20px; } .input-grid { grid-template-columns: repeat(2, 1fr); } h1 { font-size: 1.8rem; } button { font-size: 1rem; padding: 10px 20px; } #result { font-size: 1.2rem; } } @media (max-width: 480px) { .input-grid { grid-template-columns: 1fr; } h1 { font-size: 1.5rem; } }

3×3 Matrix Inverse Calculator

Enter Matrix Elements (A)

Inverse Matrix (A-1)

Enter matrix elements above.

Understanding How to Calculate the Inverse of a 3×3 Matrix

Calculating the inverse of a matrix is a fundamental operation in linear algebra with wide-ranging applications in fields like engineering, computer graphics, statistics, and solving systems of linear equations. For a 3×3 matrix, finding its inverse can be a manual or computational process. The inverse of a square matrix 'A', denoted as A-1, is a matrix such that when multiplied by 'A', it results in the identity matrix (I). That is, A * A-1 = A-1 * A = I.

A matrix has an inverse if and only if its determinant is non-zero. If the determinant is zero, the matrix is called singular, and it does not have an inverse.

Steps to Calculate the Inverse of a 3×3 Matrix

For a general 3×3 matrix A:

A = [[a11, a12, a13],
[a21, a22, a23],
[a31, a32, a33]]

The inverse A-1 can be found using the formula:

A-1 = (1 / det(A)) * adj(A)

Where:

  • det(A) is the determinant of matrix A.
  • adj(A) is the adjugate (or adjoint) of matrix A.

1. Calculate the Determinant (det(A))

The determinant of a 3×3 matrix is calculated as follows:

det(A) = a11 * (a22*a33 - a23*a32) - a12 * (a21*a33 - a23*a31) + a13 * (a21*a32 - a22*a31)

If det(A) is 0, the matrix is singular and has no inverse.

2. Calculate the Matrix of Cofactors

The cofactor Cij of an element aij is calculated by: Cij = (-1)i+j * Mij where Mij is the minor of aij (the determinant of the 2×2 matrix obtained by removing the i-th row and j-th column).

C11 = +(a22*a33 - a23*a32)
C12 = -(a21*a33 - a23*a31)
C13 = +(a21*a32 - a22*a31)
C21 = -(a12*a33 - a13*a32)
C22 = +(a11*a33 - a13*a31)
C23 = -(a11*a32 - a12*a31)
C31 = +(a12*a23 - a13*a22)
C32 = -(a11*a23 - a13*a21)
C33 = +(a11*a22 - a12*a21)

The matrix of cofactors is: C = [[C11, C12, C13],
[C21, C22, C23],
[C31, C32, C33]]

3. Find the Adjugate Matrix (adj(A))

The adjugate matrix is the transpose of the cofactor matrix:

adj(A) = CT = [[C11, C21, C31],
[C12, C22, C32],
[C13, C23, C33]]

4. Calculate the Inverse Matrix (A-1)

Finally, divide each element of the adjugate matrix by the determinant:

A-1 = (1 / det(A)) * adj(A)

Use Cases

  • Solving Systems of Linear Equations: The system AX = B can be solved as X = A-1B if A is invertible.
  • Computer Graphics: Used in transformations, projections, and modeling.
  • Robotics: For inverse kinematics problems.
  • Control Systems: Analyzing and designing control systems.
function getInputValue(id) { var input = document.getElementById(id); if (input && input.value) { return parseFloat(input.value); } return NaN; // Return NaN if input is missing or empty } function formatMatrix(matrix) { if (!matrix) return "N/A"; var formatted = ""; for (var i = 0; i < matrix.length; i++) { for (var j = 0; j < matrix[i].length; j++) { var val = matrix[i][j]; if (isNaN(val)) { formatted += " NaN "; } else { formatted += val.toFixed(6) + (j < matrix[i].length – 1 ? ", " : ""); } } formatted += (i < matrix.length – 1 ? "\n " : ""); } formatted += ""; return formatted; } function calculateInverse() { var errorMessageDiv = document.getElementById("errorMessage"); errorMessageDiv.textContent = ""; // Clear previous error messages var resultDiv = document.getElementById("result"); var matrix = []; var rows = 3; var cols = 3; var validInputs = true; for (var i = 0; i < rows; i++) { matrix[i] = []; for (var j = 0; j < cols; j++) { var val = getInputValue('a' + (i + 1) + (j + 1)); if (isNaN(val)) { validInputs = false; break; } matrix[i][j] = val; } if (!validInputs) break; } if (!validInputs) { errorMessageDiv.textContent = "Please enter valid numbers for all matrix elements."; resultDiv.innerHTML = "Enter matrix elements above."; return; } // Calculate Determinant var det = matrix[0][0] * (matrix[1][1] * matrix[2][2] – matrix[1][2] * matrix[2][1]) – matrix[0][1] * (matrix[1][0] * matrix[2][2] – matrix[1][2] * matrix[2][0]) + matrix[0][2] * (matrix[1][0] * matrix[2][1] – matrix[1][1] * matrix[2][0]); if (Math.abs(det) < 1e-10) { // Check if determinant is close to zero errorMessageDiv.textContent = "Matrix is singular (determinant is zero) and has no inverse."; resultDiv.innerHTML = "Singular Matrix"; return; } // Calculate Cofactor Matrix var cofactors = [ [0, 0, 0], [0, 0, 0], [0, 0, 0] ]; cofactors[0][0] = +(matrix[1][1] * matrix[2][2] – matrix[1][2] * matrix[2][1]); cofactors[0][1] = -(matrix[1][0] * matrix[2][2] – matrix[1][2] * matrix[2][0]); cofactors[0][2] = +(matrix[1][0] * matrix[2][1] – matrix[1][1] * matrix[2][0]); cofactors[1][0] = -(matrix[0][1] * matrix[2][2] – matrix[0][2] * matrix[2][1]); cofactors[1][1] = +(matrix[0][0] * matrix[2][2] – matrix[0][2] * matrix[2][0]); cofactors[1][2] = -(matrix[0][0] * matrix[2][1] – matrix[0][1] * matrix[2][0]); cofactors[2][0] = +(matrix[0][1] * matrix[1][2] – matrix[0][2] * matrix[1][1]); cofactors[2][1] = -(matrix[0][0] * matrix[1][2] – matrix[0][2] * matrix[1][0]); cofactors[2][2] = +(matrix[0][0] * matrix[1][1] – matrix[0][1] * matrix[1][0]); // Transpose Cofactor Matrix to get Adjugate Matrix var adjugate = [ [cofactors[0][0], cofactors[1][0], cofactors[2][0]], [cofactors[0][1], cofactors[1][1], cofactors[2][1]], [cofactors[0][2], cofactors[1][2], cofactors[2][2]] ]; // Calculate Inverse Matrix (A^-1 = (1/det) * adj(A)) var inverseMatrix = [ [0, 0, 0], [0, 0, 0], [0, 0, 0] ]; for (var i = 0; i < rows; i++) { for (var j = 0; j < cols; j++) { inverseMatrix[i][j] = (1 / det) * adjugate[i][j]; } } resultDiv.innerHTML = formatMatrix(inverseMatrix); }

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